I read what many Londoners say about Uber like they never heard about Addison Lee cabs, which existed years before, offer a nice app, professional drivers and fixed tariffs. Last time I checked, tariffs were about 20% higher than Uber. But depending on congestion, could end up cheaper. Most of my friends were praising Uber, but I was warning them that their prices are unsustainable and slightly higher easy-to-hail cab services with apps have been for years on the market largely unnoticed by hipsters ;-)
Sure you can. Nearest car is shown on the map and in my experience there was always something very close. Just choose "NOW" and it'll get booked.
Besides, comparing pre-booking, it's done properly with AL, because it sends you a confirmation right away after driver accepts. With Uber the system would simply "schedule" your booking for the time specified and then fire request up as usual. Which means you're at a mercy of nearby driver's acceptance. And you have a false expectations that the pre-booking is more timely assured.
But there are fast and frequent train services that cost less than that, so it doesn't make a huge difference whether or not Uber is an option for that route.
So get the AL to Paddington then get on the Heathrow Express. The Heathrow Express has never been very crowded in my experience, and has at least as much room for luggage as a cab does.